Kerodon

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Corollary 5.1.4.10. Let $q: X \rightarrow S$ be a morphism of simplicial sets, where $S$ is a Kan complex. The following conditions are equivalent:

$(1)$

The morphism $q$ is an isofibration.

$(2)$

The morphism $q$ is a cartesian fibration.

$(3)$

The morphism $q$ is a cocartesian fibration.

Proof. We will prove the equivalence $(1) \Leftrightarrow (2)$; the equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument. The implication $(2) \Rightarrow (1)$ is a special case of Proposition 5.1.4.8. For the converse, suppose that $q$ is an isofibration. Then $q$ is an inner fibration. To complete the proof, we must show that for every vertex $y \in X$ and every edge $\overline{e}: \overline{x} \rightarrow q(y)$ of $S$, we can write $\overline{e} = q(e)$ for some $q$-cartesian edge $e$ of $X$. Since $S$ is a Kan complex, $\overline{e}$ is an isomorphism (Proposition 1.3.6.10). Our assumption that $q$ is an isofibration then guarantees that we can write $\overline{e} = q(e)$ for some isomorphism $e: x \rightarrow y$ of $X$. The edge $e$ is automatically $q$-cartesian by virtue of Corollary 5.1.1.10. $\square$